(a) Let A be the subset of R defined by A= {1-:n€N}. Find all cluster points of A and justify your answer. (b) Let ScR and let e eR be a cluster point of S. Prove that for each e> 0 the set W:=sn(e-t.c+e) has infinitely many elements. Let (z.) be a sequence of real numbers and let B be the subset of R defined by B:= {!:n€N}. Let LE R and let f : B→R be defined by f () = a,-. Prove that lim a, = L if and only if limf(r) = L.
(a) Let A be the subset of R defined by A= {1-:n€N}. Find all cluster points of A and justify your answer. (b) Let ScR and let e eR be a cluster point of S. Prove that for each e> 0 the set W:=sn(e-t.c+e) has infinitely many elements. Let (z.) be a sequence of real numbers and let B be the subset of R defined by B:= {!:n€N}. Let LE R and let f : B→R be defined by f () = a,-. Prove that lim a, = L if and only if limf(r) = L.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(a) Let A be the subset of R defined by A= {1-:n€N}. Find all cluster points of
A and justify your answer.
(b) Let ScR and let e eR be a cluster point of S. Prove that for each e> 0 the set
W:=sn(e-t.c+e) has infinitely many elements.
Let (z.) be a sequence of real numbers and let B be the subset of R defined by B:=
{!:n€N}. Let LE R and let f : B→R be defined by f () = a,-. Prove that lim a, = L if
and only if limf(r) = L.
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